Integral homology -spheres and the Johnson filtration
Author:
Wolfgang Pitsch
Journal:
Trans. Amer. Math. Soc. 360 (2008), 2825-2847
MSC (2000):
Primary 57M99; Secondary 20F38, 20F12
DOI:
https://doi.org/10.1090/S0002-9947-08-04208-6
Published electronically:
January 4, 2008
MathSciNet review:
2379777
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: The mapping class group of an oriented surface of genus
with one boundary component has a natural decreasing filtration
, where
is the kernel of the action of
on the
nilpotent quotient of
. Using a tree Lie algebra approximating the graded Lie algebra
we prove that any integral homology sphere of dimension
has for some
a Heegaard decomposition of the form
, where
and
is such that
. This proves a conjecture due to S. Morita and shows that the ``core'' of the Casson invariant is indeed the Casson invariant.
- 1. Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423–472. MR 1318886, https://doi.org/10.1016/0040-9383(95)93237-2
- 2. Robert Craggs, A new proof of the Reidemeister-Singer theorem on stable equivalence of Heegaard splittings, Proc. Amer. Math. Soc. 57 (1976), no. 1, 143–147. MR 410749, https://doi.org/10.1090/S0002-9939-1976-0410749-9
- 3. Stavros Garoufalidis and Jerome Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math., vol. 73, Amer. Math. Soc., Providence, RI, 2005, pp. 173–203. MR 2131016, https://doi.org/10.1090/pspum/073/2131016
- 4. H. B. Griffiths, Automorphisms of a 3-dimensional handlebody, Abh. Math. Sem. Univ. Hamburg 26 (1963/64), 191–210. MR 159313, https://doi.org/10.1007/BF02992786
- 5. Nathan Habegger and Gregor Masbaum, The Kontsevich integral and Milnor’s invariants, Topology 39 (2000), no. 6, 1253–1289. MR 1783857, https://doi.org/10.1016/S0040-9383(99)00041-5
- 6. Nathan Habegger and Wolfgang Pitsch, Tree level Lie algebra structures of perturbative invariants, J. Knot Theory Ramifications 12 (2003), no. 3, 333–345. MR 1983089, https://doi.org/10.1142/S0218216503002494
- 7. Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
- 8. Dennis Johnson, The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves, Topology 24 (1985), no. 2, 113–126. MR 793178, https://doi.org/10.1016/0040-9383(85)90049-7
- 9.
Jerome
Levine, Addendum and correction to: “Homology cylinders: an
enlargement of the mapping class group” [Algebr. Geom. Topol. 1
(2001), 243–270; MR1823501 (2002m:57020)], Algebr. Geom. Topol.
2 (2002), 1197–1204. MR
1943338, https://doi.org/10.2140/agt.2002.2.1197
Jerome Levine, Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001), 243–270. MR 1823501, https://doi.org/10.2140/agt.2001.1.243 - 10. Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, 2nd ed., Dover Publications, Inc., Mineola, NY, 2004. Presentations of groups in terms of generators and relations. MR 2109550
- 11. Shigeyuki Morita, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles. I, Topology 28 (1989), no. 3, 305–323. MR 1014464, https://doi.org/10.1016/0040-9383(89)90011-6
- 12. Shigeyuki Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993), no. 3, 699–726. MR 1224104, https://doi.org/10.1215/S0012-7094-93-07017-2
- 13. J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen i, Acta Math 50 (1927), 189-358.
- 14. -, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen II, Acta Math 53 (1929), 1-76.
- 15. -, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen III, Acta Math 58 (1931), 87-167.
- 16. Wolfgang Pitsch, Une construction intrinsèque du cœur de l’invariant de Casson, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 6, 1741–1761 (French, with English and French summaries). MR 1871288
- 17. Wolfgang Pitsch, Extensions verselles et automorphismes des groupes nilpotents libres, J. Algebra 249 (2002), no. 2, 512–527 (French, with French summary). MR 1901170, https://doi.org/10.1006/jabr.2001.9077
- 18. K. Reidemeister, Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg (1933), 189-194.
- 19. James Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933), no. 1, 88–111. MR 1501673, https://doi.org/10.1090/S0002-9947-1933-1501673-5
- 20. Shin’ichi Suzuki, On homeomorphisms of a 3-dimensional handlebody, Canadian J. Math. 29 (1977), no. 1, 111–124. MR 433433, https://doi.org/10.4153/CJM-1977-011-1
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57M99, 20F38, 20F12
Retrieve articles in all journals with MSC (2000): 57M99, 20F38, 20F12
Additional Information
Wolfgang Pitsch
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
Email:
pitsch@mat.uab.es
DOI:
https://doi.org/10.1090/S0002-9947-08-04208-6
Received by editor(s):
November 7, 2005
Published electronically:
January 4, 2008
Additional Notes:
The author was supported by MEC grant MTM2004-06686 and by the program Ramón y Cajal, MEC, Spain
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.